# Group having 3 proper subgroups

#### Chandru1

What can we say about a group which has exactly 3 proper subgroups?

#### tonio

What can we say about a group which has exactly 3 proper subgroups?

Since any non-trivial group always has two trivial subgroups (the trivial one and the whole group), we're looking for a group with one single non-trivial subgroup...hint: how many primes can divide the group's order?

Tonio

#### Swlabr

Since any non-trivial group always has two trivial subgroups (the trivial one and the whole group), we're looking for a group with one single non-trivial subgroup...hint: how many primes can divide the group's order?

Tonio
No - proper means it is properly contained in it. That is to say, $$\displaystyle H \lneq G$$. So we are looking for a group with precisely two non-trivial subgroups (although I think your hint is still the way to go).

#### TheArtofSymmetry

What can we say about a group which has exactly 3 proper subgroups?
As tonio said, a cyclic group of order $$\displaystyle p^3$$ can be one of examples, where p is a prime number.

See my previous post here .

#### Swlabr

As tonio said, a cyclic group of order $$\displaystyle p^3$$ can be one of examples, where p is a prime number.

See my previous post here .
As a warning to the OP, this does not classify them all. Another example would be $$\displaystyle V_4$$, the klein 4-group (or, more generally, the cross-product of two groups of prime order).

#### tonio

No - proper means it is properly contained in it. That is to say, $$\displaystyle H \lneq G$$. So we are looking for a group with precisely two non-trivial subgroups (although I think your hint is still the way to go).

Yes. I oversaw the word "proper" in the OP, but still the hint remains...though nevertheless there are OTHER examples: for example, a cyclic group of order 10...

Tonio